Liquidity-Based Audit of Algorithmic Trading Strategies

Liquidity-Based Audit of Algorithmic Trading Strategies

introduction

Market microstructure has long classified trading activity by its informational role: an informed trader demands liquidity by trading in the direction of private information, while a market maker supplies liquidity by absorbing that order flow and earning the spread in compensation Kyle1985,Glosten1985. This classification is typically recovered from the data the classifier requires: signed order flow, quote revisions, or the sequential-trade structure of the market. The classification is harder to apply to an algorithmic strategy whose internal logic is unobservable. However, the signals or optimization problems generating the decisions of a typical quantitative fund are not visible, even though the trades and reported positions may be available.

This paper shows that the liquidity role of such a strategy (consumer or provider) can be recovered from realized portfolio costs and trade decisions alone, without observing quotes, order flow, or any other microstructure-specific signal. The key observation is that a strategy's contribution to multi-period performance decomposes exactly into a sum of per-period covariances between the realized cost vector and the strategy's decision (Theorem (ref)). Furthermore, the sign of this covariance statistic is a model-free liquidity classifier: positive covariance means the strategy systematically buys when costs are rising and sells when costs are falling, while negative covariance means the reverse. Buying when asset costs (prices) are rising and selling when prices are falling is the behavior of a successful quantitative fund. The reverse is the defining behavior of a liquidity supplier (Proposition (ref)).

This single observation has three further consequences that are the substance of the paper. First, under an AR(1) cost process (the standard reduced-form model for short-horizon return reversal), the same covariance statistic admits a closed form in terms of the roll1984simple implied bid-ask spread. The auto-covariance correction to the covariance sum equals $\alpha s^2$, the product of strategy size and the squared effective spread (Theorem (ref)). This gives a spread estimator that is a byproduct of the same trajectory data used for the liquidity classification, at no additional computational cost.

Second, allowing the strategy's own trades to move prices shows that covariance regret and price-impact regret are additively separable (Theorem (ref)). This is the endogenous price impact in the spirit of Kyle1985 and almgren2001optimal. The additive separation allows the two channels to be estimated independently from the same data.

Third, aggregating across $N$ strategies under a common-factor cost structure yields a market-wide liquidity-balance condition: when liquidity demand and supply fail to net to zero across agents. Prices must adjust to clear, and the resulting welfare loss scales as $N^2$ in the strategy-correlation coefficient (Corollary (ref)). The result is a closed-form statement of the familiar fire-sale mechanism in which correlated deleveraging overwhelms available liquidity.

We calibrate the framework to CRSP daily equity data from 2016 through 2025 ($n=21{,}183{,}373$ stock-days). The implied Roll spread recovered from the trajectory estimator tracks realized illiquidity through two distinct episodes: it roughly triples during the COVID-19 liquidity crisis of Q2 2020 (from $0.90\%$ to $2.52\%$ daily), and it collapses to its sample minimum during the 2022 rate-shock episode, when persistent common-factor repricing rather than microstructure friction dominates short-horizon return dynamics. Both patterns are consistent with the spread acting as a real-time proxy for the depth of the market rather than an estimation artifact.

\paragraph{Related work.} The liquidity-classification result connects most directly to the sequential-trade and strategic-trading literature. Kyle1985 and Glosten1985 establish the informed-trader/market-maker dichotomy that this paper recovers from trajectory data rather than order-flow data. roll1984simple provides the implied-spread estimator that this paper re-derives as a regret correction. almgren2001optimal and gatheral2010no provide the price-impact and market-impact models extended in Section (ref). Hasbrouck1991 and the broader VAR-based price-impact literature offer an alternative, order-flow-based route to the same quantities that this paper recovers from realized costs and decisions alone, a comparison we return to in Section (ref). The covariance decomposition itself extends the single-period identity of aldridge2026regret to the full multi-period stochastic dynamic programming setting and is structurally related to the elmachtoub2022smart predict-then-optimize framework and to the regret criterion in online convex optimization Zinkevich2003,Hazan2016. The $N$-agent fire-sale extension connects to the market-wide liquidity-spiral literature ShleiferVishny1992 and to random matrix theory BaiSilverstein2010 for the systemic-fragility threshold of Section (ref).

\paragraph{Roadmap.} Section (ref) states the liquidity classification result in its simplest form, immediately after the minimal notation required to state it. Section (ref) develops the full multi-period stochastic dynamic programming setup and proves the exact covariance decomposition (Theorem (ref)) that underlies everything that follows. Section (ref) gives a Bellman recursion for the covariance regret functional. Section (ref) characterizes the policy bias that arises for strategies (e.g., momentum) that violate the mean-unbiasedness condition. Section (ref) derives the closed-form auto-covariance correction under an AR(1) cost process. Section (ref) returns to liquidity classification in full: the Roll-spread identity, endogenous price impact, the aggregate liquidity-balance condition and fire-sale extension, and the empirical calibration of implied spreads against the CRSP sample. Sections that follow develop the trajectory estimator's asymptotic properties, additional empirical results, and an $N$-agent systemic extension.

A First Look at Liquidity Classification

Before developing the full multi-period framework, we state the paper's central classification result in its simplest form, since it requires almost none of the machinery that follows.

Consider a single trading strategy that, at each period $t$, faces a cost vector $c_t\in\mathbb{R}^d$ (e.g.\ a vector of asset return innovations or factor realizations) and chooses a position $\hat{\pi}_t(c_t)\in\mathbb{R}^n$ as a linear function of that cost: $\hat{\pi}_t(c_t) = Bc_t + b$ for some matrix $B\in\mathbb{R}^{n\times d}$. Let $\Sigma_c=\operatorname{Cov}(c_t,c_t)$ be the cost covariance matrix. No further structure is needed for the result below; the full stochastic dynamic programming setup, including the multi-period regret identity that gives this statistic its performance interpretation, is developed starting in Section (ref).

The proof follows directly from the bilinearity of covariance and the cyclic property of trace. The proof appears in full as Proposition (ref) in Section (ref), along with the connection to Kyle1985 and the closed-form Roll-spread identity that follows from the same statistic under an AR(1) cost process. The point to take from this first look is that the sign of a single, easily computed statistic classifies the strategy's liquidity role exactly. This requires only the observed trajectory $\{(c_t,\hat{\pi}_t(c_t))\}_{t=1}^T$ and no knowledge of $B$, the strategy's signal, or its optimization problem.

Sections (ref) through (ref) develop the multi-period regret framework in which this statistic also functions as a performance audit metric. Section (ref) returns to the liquidity interpretation with the full set of results, including the Roll-spread identity, price impact, and the $N$-agent aggregate liquidity-balance condition previewed above.

Setup: Multi-Period Stochastic Dynamic Program

Model

At each period $t \in \{1,\ldots,T\}$ a state $s_t \in \mathcal{S} \subseteq \mathbb{R}^p$ summarizes relevant history, a cost vector $c_t \in \mathcal{C} \subseteq \mathbb{R}^d$ is drawn from a distribution depending on $s_t$, and a decision $z_t \in \mathcal{Z} \subseteq \mathbb{R}^n$ is chosen by a policy $\hat{\pi}_t(s_t, c_t)$ after observing $(s_t, c_t)$, realizing instantaneous cost $c_t^\top z_t$. The state evolves as $s_{t+1} = f(s_t, z_t, \varepsilon_{t+1})$.

Let $\mathcal{F}_t = \sigma(s_1,c_1,\ldots,s_t,c_t)$ be the natural filtration and $\mathbb{E}_t[\,\cdot\,] = \mathbb{E}[\,\cdot\mid \mathcal{F}_t]$. Write $\mu_t = \mathbb{E}[c_t\mid s_t]$ and $\Sigma_t = \operatorname{Cov}(c_t,c_t\mid s_t)$.

Main decomposition

Bellman Recursion for the Covariance Regret Functional

When Does the Identity Fail? Bias Analysis for Time-Varying Policies

We characterize when the exact identity $\operatorname{Regret}^{(T)} = \sum_t \operatorname{Cov}(c_t,\hat{\pi}_t)$ fails and derive the resulting bias in closed form.

Markovian Cost Processes: Auto-Covariance Correction

When costs follow an AR(1) process, the per-period covariances $\operatorname{Cov}(c_t,\hat{\pi}_t)$ are no longer independent across time. We derive the resulting cross-period correction in closed form.

The Audit Metric as a Measure of Liquidity Provision and Consumption

The covariance identity of Theorem (ref) is usually read as a performance diagnostic: $\operatorname{Cov}(c_t,\hat{\pi}_t)>0$ means the strategy incurs positive expected regret, and a compliance officer should care about how large that covariance is. This section shows that the same statistic has a second, independent interpretation: it is a model-free measure of net market liquidity demand. A positive covariance identifies the strategy as a liquidity consumer; a negative covariance identifies it as a liquidity provider; and the sign alone, observable from inputs and outputs without access to the internal optimizer, determines the role. This reinterpretation connects the audit framework to a large body of market-microstructure theory, yields a closed-form link between the AR(1) correction of Section (ref) and the Roll roll1984simple implied bid-ask spread, and provides an aggregate liquidity-balance condition whose violation characterizes the fire-sale episodes studied in Section (ref).

Liquidity Classification via the Audit Metric

We formalize the connection between the covariance statistic and market-microstructure roles.

The definition is intentionally model-free: it requires only the observable sequence $\{(c_t,\hat{\pi}_t(c_t))\}$ and imposes no assumptions about whether the policy is momentum-based, contrarian, or otherwise. The next result shows that for linear policies, the sign of the audit metric is equivalent to the sign of the leading eigenvalue of the policy matrix $B$.

The Roll Spread and the AR(1) Correction

The connection between the audit metric and market liquidity deepens in the AR(1) setting of Section (ref). We show that the auto-covariance correction of Theorem (ref) equals a closed-form function of the roll1984simple implied bid-ask spread, turning the regret decomposition into a direct measure of liquidity-provision income.

Theorem (ref) also provides a simple empirical formula for translating the AR(1) coefficient directly into a spread estimate.

Endogenous Price Impact and Effective Regret

The Roll model captures bid-ask bounce but treats prices as exogenous. We now allow the agent's own trades to move prices, introducing endogenous price impact in the spirit of Kyle1985 and almgren2001optimal.

Aggregate Liquidity Balance and Market Fragility

The single-agent results aggregate cleanly to a market-wide liquidity balance condition: for prices to clear, the net demand for liquidity across all agents must sum to zero. When it does not, prices must adjust until it does, producing the market impact that generates fire sales.

Empirical Calibration: Spread, Impact, and Fragility

Table (ref) reports the Roll implied spread $\hat{s}$ and its connection to the AR(1) correction and fragility multiplier for the 2016--2025 CRSP sample, using equation (ref). For each year we compute $\hat{s}=\hat{\sigma}_c\sqrt{|\hat{\rho}|}$ where $\hat{\sigma}_c$ and $\hat{\rho}$ are taken from Table (ref).

Three observations stand out. First, $\hat{s}$ and the bias percentage move nearly in lockstep across years: the Roll spread is a sufficient statistic for the size of the AR(1) correction, as predicted by Theorem (ref). Second, the COVID crisis Q2 2020 row shows that a tripling of the Roll spread ($0.90\%$ in 2016 to $2.52\%$ in crisis) corresponds to a tripling of the fragility multiplier and a four-fold increase in the bias percentage (compared to a typical pre-COVID year). Third, 2022 is exceptional in both dimensions: the Roll spread ($0.25\%$/day) and fragility multiplier ($1.4\times$) are the lowest in the sample, confirming the "signal-collapse" reading of Section (ref): the rate shock compressed bid-ask spreads and strategy correlations simultaneously, producing a regime where neither the audit metric nor the fragility multiplier provides useful guidance. The structural shift in the cost process $c_t$ is the operative force, not microstructure.

Financial Applications

Rolling-window portfolio rebalancing

A manager rebalancing daily using a $w$-day rolling window applies the policy $\hat{\pi}_t(c_t)=\hat{\Sigma}_t^{-1}\hat{\mu}_t / (\mathbf{1}^\top\hat{\Sigma}_t^{-1}\hat{\mu}_t)$, where $\hat{\mu}_t$ and $\hat{\Sigma}_t$ are sample moments from the most recent $w$ observations.

For i.i.d.\ returns with $d$ assets and $w>d+1$, the per-period policy bias satisfies $\|b_t\| = O(d/w)$ and the raw covariance sum approximates true regret with relative error $O(d/w)$, vanishing as $w/d\to\infty$ ledoit2004well. Three regimes determine which estimator to use: $w\geq 10d$ (raw covariance sum reliable); $d<w<10d$ (use bias-corrected estimator, Proposition (ref)); $w\leq d$ ($\hat{\Sigma}_t$ rank-deficient, regularization required before any regret estimate is meaningful). For a typical institutional setting ($T=252$, $d=100$, $w=60$), the portfolio falls in the third regime and $\hat{\Sigma}_t$ must be regularized (e.g., Ledoit-Wolf shrinkage ledoit2004well or factor-model reduction to $d'\leq 60$ latent factors) before applying the decomposition.

Mean-reverting execution costs

In algorithmic execution, market impact is mean-reverting ($c_t=\rho c_{t-1}+\varepsilon_t$, $\rho\in(-1,0)$); the execution policy allocates $\hat{\pi}_t(c_t)=\alpha c_t$ to exploit temporary mispricings.

Multi-Period Regret Estimation from Observed Trajectories

The trajectory covariance estimator

Given a single observed trajectory $\{(c_t,\hat{\pi}_t(c_t))\}_{t=1}^T$, the natural estimator of the sum-of-covariances is

Confidence intervals

A $(1-\alpha)$ confidence interval for total regret is

where $\hat{\sigma}_{\mathrm{LRV}}$ is the square root of any consistent HAC variance estimator (e.g., Newey-West with $h=\lfloor T^{1/3}\rfloor$ lags).

Empirical Analysis: Regret Decomposition Across Financial Fragility Episodes

Financial fragility does not distribute evenly across time. Mean-reversion, momentum, and the welfare gap between them wax and wane with market-wide liquidity conditions. This section examines the CRSP-based data evidence through that lens. We show that the AR(1) autocorrelation coefficient $\hat{\rho}$ is not merely a statistical nuisance parameter to be corrected for: it is a real-time fragility signal that compresses toward zero when market microstructure breaks down, spikes in absolute value during liquidity crises, and, through the bias-corrected regret decomposition of Theorem (ref), produces a welfare-loss measure that anticipates strategy failure before Sharpe ratios do.

We organize the empirical evidence around three episodes that the 2016--2025 sample straddles: (i) a sequence of normal-market years (2016--2019, 2021, 2023--2025) in which mean-reversion is consistent and can be readily monetized; (ii) the COVID-19 liquidity crisis of 2020, in which a brief period of momentum dominance interrupts sustained reversion and the fragility multiplier of Section (ref) nearly doubles; and (iii) the rate-shock episode of 2022, in which the Federal Reserve's fastest hiking cycle in four decades collapses $\hat{\rho}$ to near zero, disabling both mean-reversion and momentum and producing the only calendar year in the sample where all four strategies post negative or near-zero risk-adjusted returns.

Data and Estimation Strategy

We use CRSP daily stock returns (WRDS DSI) from January 4, 2016, through December 31, 2025, comprising $n = 21{,}183{,}373$ stock-day observations ($\bar{\mu} = 0.041\%$/day, $\bar{\sigma} = 4.446\%$/day). The momentum signal is a binary price-direction indicator: buy if $R_{t-1}>0$, sell otherwise. For each calendar year, we estimate the pooled AR(1) coefficient

by OLS with Newey-West HAC standard errors (bandwidth $h=\lfloor T^{1/3}\rfloor$). All ADF tests reject the unit root at $p<0.0001$. We compute the regret decomposition of Theorem (ref) at horizon $T=252$ and report annualized Sharpe ratios on a 21-day rolling window for four strategies: momentum (MOM), reversion (REV), bias-corrected reversion (BC), and minimum-variance (MV). NBER business-cycle classifications partition the 2020 sample into Expansion (January--February), Contraction (March), and Crisis (April--June).

Normal-Market Baseline (2016--2019, 2021, 2023--2025)

Outside the two fragility episodes, the data are remarkably uniform. AR(1) coefficients lie in the range $\hat{\rho}\in[-0.054,-0.013]$ across eight calendar years, all significant at $p<0.05$ or better. The AR(1) auto-covariance correction consistently reduces the raw covariance sum, with relative bias ranging from $-1.32\%$ (2024) to $-4.87\%$ (2016). The reversion strategy outperforms momentum in every normal-market year, with full-sample Sharpe ratios of $0.130$ and $0.023$ respectively.

These results confirm the diagnostic-sufficiency conclusion of Section (ref): monitoring the per-period sample covariance $\widehat{\operatorname{Cov}}(c_t,\hat{\pi}_t)$ is sufficient to track multi-period performance across the full normal-market subsample. The bias correction matters in magnitude ($1\%$--$5\%$ of true regret), but not in sign or direction: in stable conditions, the estimator is conservative in the right direction, and no year crosses the threshold at which the raw sum materially misleads a compliance officer.

The minimum-variance strategy exhibits the lowest volatility throughout, consistent with Assumption (ref) holding exactly for mean-unbiased policies (bias correction $\approx 0$) and with the known variance-reduction properties of Ledoit-Wolf shrinkage ledoit2004well.

Financial Fragility Episode I: COVID-19 Liquidity Crisis (2020)

The year 2020 is the single outlier in a decade of consistent mean-reversion, and it is an outlier precisely because it contains a textbook financial fragility episode.

\paragraph{Anatomy of the reversal.} Table (ref) decomposes the 2020 annual AR(1) estimate ($\hat{\rho}=-0.050$, SE $= 0.004$) into three NBER sub-periods. In the expansion phase (January--February, $n=468{,}004$ stock-days), mean-reversion operates normally at $\hat{\rho}=-0.0146$. As pandemic news and repo-market stress materialize in March (contraction, $n=250{,}035$), $\hat{\rho}$ doubles in absolute value to $-0.0270$. During the April--June crisis phase ($n=219{,}742$), $\hat{\rho}$ spikes to $-0.1436$. This is a tenfold amplification relative to expansion and the largest single-quarter reversal in the full 2016--2025 sample.

\paragraph{The momentum anomaly and its interpretation.} The full-year 2020 Sharpe ratio for momentum ($0.261$) exceeds that of reversion ($0.217$), the only such inversion in the sample. The mechanism is temporal: March 2020 saw a directional, cross-asset collapse that rewarded trend-following strategies precisely because prices were not mean-reverting. The prices were being set by forced liquidation and margin calls rather than by informed trading. This is the fire-sale mechanism formalized in Remark (ref): aggregate positions became correlated, price impact overwhelmed idiosyncratic reversion, and the mean-reversion signal went dark for six to eight weeks.

Consistent with this interpretation, the bias correction identifies the anomaly in advance. The BC Sharpe ratio for 2020 ($0.232$) exceeds the raw REV Sharpe ($0.217$): the bias-corrected estimator correctly down-weights the reversion signal during the contraction phase, when $|\hat{\rho}|$ is rising but has not yet dominated. The BC strategy, by scaling the reversion allocation by $14\times$ the bias correction (see Table (ref) note d), reduces exposure as the fragility multiplier $M(N,\rho_{\mathrm{sys}})$ rises from $18.8\times$ to $32.7\times$ between January and March.

\paragraph{Regret decomposition during crisis.} The true annual regret for 2020 is $7{,}859$, against a raw covariance sum of $7{,}503$: the AR(1) correction accounts for $-357$ units, the largest absolute correction in the sample. As $\hat{\rho}$ spikes to $-0.144$ in the crisis quarter, the auto-covariance correction of Theorem (ref) grows proportionally to $|\hat{\rho}|$, generating a correction that is $4.54\%$ of true regret at the annual horizon but would be $\sim 15\%$--$20\%$ at the quarterly ($T=63$) horizon when computed over the crisis sub-period alone. This is a direct empirical illustration of Corollary (ref): in mean-reverting markets, the raw covariance sum underestimates true regret, and the bias grows as the degree of mean-reversion intensifies. An auditor relying solely on the raw covariance sum during Q2 2020 would have systematically understated strategy-specific welfare losses by one-fifth, precisely the period when regulatory attention was most acute.

Financial Fragility Episode II: Rate-Shock Breakdown (2022)

The 2022 fragility episode is structurally distinct from 2020. Where COVID produced a liquidity-driven spike in mean-reversion, the Fed's 425-basis-point hiking cycle produced a structural collapse of the reversion signal.

\paragraph{Near-zero autocorrelation.} The 2022 annual AR(1) estimate is $\hat{\rho}=-0.0036$ (SE $=0.0028$, $p<0.10$), the smallest absolute value in the decade and an order of magnitude below the 2016--2021 average of $\hat{\rho}\approx -0.045$. This near-zero autocorrelation has a natural structural explanation: rising interest rates repriced virtually every asset class simultaneously and persistently across 2022, generating a sustained directional drift that overwhelmed the short-horizon reversion dynamics that dominate in mean-stationary environments. Formally, the AR(1) coefficient $\rho$ measures the sign persistence of idiosyncratic price innovations; when a common factor (rates) drives persistent cross-sectional co-movement, the idiosyncratic component $\hat{\rho}$ compresses toward zero.

\paragraph{Strategy failure.} All four strategies underperform in 2022. The MOM Sharpe ratio is $-0.028$; REV posts $-0.102$; BC $-0.104$; MV $+0.062$. Notably, the bias correction provides no rescue in this episode: BC ($-0.104$) is marginally worse than raw REV ($-0.102$) because near-zero $\hat{\rho}$ produces a near-zero bias correction term $\bar{c}^{\top}b_t$ in equation (ref), leaving the allocator fully exposed to the directional rate shock with no compensating upward revision.

This is not a failure of the estimator, but a correct signal. When $\hat{\rho}\to 0$, the system has exited the mean-reverting regime entirely; the regret decomposition faithfully reports that no reversion-based strategy can generate positive covariance-adjusted returns. The minimum-variance portfolio, which is regime-agnostic by construction (it targets $\operatorname{Cov}(c_t,\hat{\pi}_t)=0$ by Assumption (ref)), is the only strategy that survives 2022 with a positive Sharpe ratio, providing a natural "fragility hedge" benchmark consistent with the zero-bias prediction of Theorem (ref).

\paragraph{Regret bias in 2022.} The raw covariance sum is $4{,}288$ and the AR(1) correction is merely $-15$, yielding a relative bias of $-0.35\%$. This is the smallest bias in the sample, reflecting the near-zero $\hat{\rho}$. This means that in 2022, the raw covariance sum is essentially a sufficient statistic for true regret: the absence of autocorrelation removes the need for any dynamic correction. Paradoxically, the year in which the regret decomposition is most straightforward to apply is also the year in which no strategy benefits from it, because the regime that generates exploitable covariance structure has temporarily ceased to operate.

The Regret Decomposition as a Fragility Diagnostic

Taken together, the three empirical regimes, namely normal-market, COVID crisis, and rate-shock breakdown, reveal a consistent pattern that we formalize as follows.

\paragraph{$\hat{\rho}$ as a regime indicator.} Define three regimes by the value of the rolling AR(1) coefficient:

\paragraph{Early-warning properties.} The bias-corrected estimator (ref) functions as an early-warning signal in regime (ii). In the 2020 contraction phase (March), $|\hat{\rho}|$ rises from $0.015$ to $0.027$ and the BC strategy reduces its reversion allocation, posting a Sharpe ratio of $0.232$ versus $0.217$ for raw REV over the full year. This $0.015$ Sharpe improvement is small for normal markets, but is equivalent to avoiding approximately one-quarter of the under-performance that an uncorrected reversion strategy accumulates during the March contraction before mean-reversion reasserts in April.

In regime (iii), by contrast, the bias correction correctly signals its own irrelevance: $\hat{\rho}\approx 0$ means the correction term is negligible, and no dynamic adjustment to the reversion allocation can rescue a strategy from a structural regime shift. A compliance officer using $\hat{F}_T$ (Definition (ref)) alongside $\hat{\rho}$ would observe both indicators simultaneously: rising $\hat{F}_T$ warns of fragility-spike risk (regime ii); $\hat{\rho}\to 0$ warns of signal-collapse risk (regime iii). These are complementary, not redundant, fragility indicators.

\paragraph{Horizon invariance.} At horizons $T\in\{21,63,126\}$, the relative bias $\Delta_T / \mathrm{Regret}^{(T)}_{\mathrm{true}}$ is horizon-invariant within each calendar year (range: $-0.34\%$ to $-5.12\%$), as predicted by Corollary (ref). This means the fragility diagnostic is equally informative at monthly, quarterly, and semi-annual review horizons, satisfying the "forward-looking" mandate of SR 11-7 across all standard regulatory time-scales.

Robustness

Five confounds bias $|\hat{\rho}|$ upward: (i) size/SMB effects jegadeesh2025shortterm; (ii) bid-ask bounce roll1984simple; (iii) volatility clustering engle1982autoregressive; (iv) cross-sectional versus time-series momentum mamais2025explaining; and (v) liquidity-provision dynamics nagel2012evaporating. All regret figures in Tables (ref) and (ref) are therefore conservative upper bounds on the welfare loss attributable to strategy mis-specification.

The primary directions for resolving these confounds are Fama-French five-factor residuals fama2015five and roll1984simple bid-ask correction. Both are feasible on the CRSP/WRDS data used here and are identified as the main extensions for follow-on empirical work. Importantly, the theoretical decompositions of Theorems (ref), (ref), and the systemic extension of Theorem (ref) hold exactly under Assumption (ref) regardless of the empirical source of $\hat{\rho}$; reducing the upward bias in $|\hat{\rho}|$ would tighten the reported regret bounds, not overturn the fragility-episode narrative.

The regime-switching interpretation of Section (ref) is robust to the confounds in a further sense: all five sources of upward bias in $|\hat{\rho}|$ are persistent across years, yet the regime variation we document (near-zero in 2022, spiking in Q2 2020) is orthogonal to any slowly-varying confound. Bid-ask bounce, size effects, and volatility clustering do not selectively disappear in 2022 or spike in April--June 2020. The rate-shock and COVID fragility patterns survive any correction that is approximately stationary across the sample.

Systemic Risk Extension: Correlated AI Strategies and Financial Fragility

The preceding analysis audits a single algorithmic agent in isolation. This section shows why that is insufficient for financial stability purposes. When $N$ agents independently satisfy the individual audit criterion of Theorem (ref), the aggregate economy can nonetheless exhibit unbounded fragility. The mechanism is strategy correlation: policies that respond to common cost factors create a quadratic amplification of aggregate regret that no per-agent audit detects.

Setup: $N$-Agent Economy with Common Factors

Let $N$ agents operate simultaneously. Agent $i\in\{1,\ldots,N\}$ faces cost vector $c_t^i\in\mathbb{R}^d$ and chooses allocation $z_t^i\in\mathbb{R}^n$ under policy $\hat{\pi}_t^i$. Costs share a common-factor structure.

Aggregate Regret Decomposition

Theorem (ref) decomposes aggregate regret into two terms: a sum of $N$ individual regrets. Each individual regret is bounded by the per-agent audit. A cross-agent systemic term that grows as $O(N^2)$ in the strategy-correlation matrix.

The Fragility Multiplier and Phase Transition

The cross-agent term in (ref) depends on how similar agents' strategies are. We characterize this via a scalar fragility multiplier.

The transition from regime (i) to (iii) is continuous in $\rho_{\mathrm{sys}}$. We identify a natural critical threshold using random matrix theory.

An Observable Systemic Fragility Index

The regulator does not observe $\rho_{\mathrm{sys}}$ directly, but observes $\{(c_t^i,\hat{\pi}_t^i(c_t^i))\}$ for all $i$ and $t$. We construct a consistent estimator of the fragility multiplier and derive a confidence interval for aggregate regret.

Empirical Calibration: CRSP 2016--2025

To calibrate the fragility multiplier for realistic parameter values, we use the AR(1) estimates of Table (ref) to bound $\rho_{\mathrm{sys}}$. The momentum policy $\hat{\pi}_t(c_t)=Bc_t$ with $B=\alpha I$ and the common Fama--French market factor $f_t=r_{m,t}-r_{f,t}$ imply

where $R^2_{\mathrm{mkt}}$ is the cross-sectional $R^2$ of the market-factor regression. Using the CRSP 2016--2025 data of Section (ref), $R^2_{\mathrm{mkt}}$ averages $0.27$ in normal years but rises to $0.61$ during the COVID crisis period ($\hat{\rho}=-0.144$), reflecting the sharp co-movement of individual stocks with the aggregate market during stress. Table (ref) reports the implied fragility multipliers.

Two conclusions are immediate. First, even under normal conditions, aggregate regret for 100 correlated AI strategies is approximately 27 times the individual regret. Second, during crisis episodes, $\rho_{\mathrm{sys}}$ nearly doubles, pushing $M$ to $60$ for $N=100$ institutions. This represents a form of liquidity spiral: stress raises correlation, which raises the fragility multiplier, which raises aggregate welfare loss, which deepens stress. The systemic audit framework of Theorem (ref) is designed to detect the onset of this spiral before it fully develops.

Policy Implications for Systemic AI Regulation

Three policy implications follow from Corollary (ref) and Theorem (ref).

\paragraph{Aggregate position limits.} If $\hat{F}_T$ exceeds the Marchenko--Pastur threshold of Proposition (ref), a regulator can cap individual position sizes at \[ z_{\max} \;=\; \frac{\varepsilon}{N\cdot\rho_{\mathrm{sys}}\cdot\|\bar{c}\|} \] to maintain aggregate regret below a target $\varepsilon$. This is the systemic analog of single-firm leverage limits, derived here from first principles via the regret decomposition.

\paragraph{Strategy diversity requirements.} Minimizing aggregate regret subject to fixed total capital is equivalent to minimizing $M(N,\rho_{\mathrm{sys}})$, which is achieved by minimizing $\rho_{\mathrm{sys}}$. This provides a formal welfare justification for regulatory requirements that AI-driven strategies use distinct factor-loading matrices $\Lambda_i$. These are analogous to capital diversification rules, but applied to algorithmic model design rather than balance-sheet composition.

\paragraph{Timing of intervention.} Because $\hat{F}_T$ is updated online in $O(N^2\cdot n)$ per day (Remark (ref)), the regulator observes the fragility multiplier in near-real time. A rising $\hat{F}_T$ preceding a volatility spike is an early warning of imminent systemic stress, complementing standard VaR-based indicators and satisfying the "forward-looking" mandate of SR 11-7.

Connection to Coordination Failures and Runs

The fragility multiplier provides a formal link between the regret framework and classic models of bank runs and coordination failures. Under perfect correlation ($\rho_{\mathrm{sys}}=1$), equation (ref) collapses to

This is the payoff structure of a coordination game: each agent individually minimizes regret, but collectively they produce quadratic aggregate loss, as in a prisoner's dilemma. The "run equilibrium" arises when a common cost shock causes all agents to simultaneously de-risk. De-risking becomes a policy that individually reduces each agent's $\operatorname{Cov}(c_t^i,\hat{\pi}_t^i)$ but, in aggregate, drains market liquidity and raises $c_t^i$ for all agents through price impact, precisely the mechanism of Corollary (ref). Equation (ref) therefore quantifies the welfare loss from the run equilibrium relative to the planner's optimum ($\rho_{\mathrm{sys}}=0$) in a closed form that is empirically estimable from observable data alone.

This connection is more than analogical. In the two-period version of the model, the game played by $N$ agents is a linear-quadratic game with price-impact externality, and the Nash equilibrium regret is exactly $M(N,\rho_{\mathrm{sys}})$ times the social optimum. The full game-theoretic mixed-strategy equilibria arise when agents have private information about the factor loading $\Lambda_i$. Deriving these equilibria is an important direction for future work.

Conclusion

A strategy's liquidity role is written into the covariance between its trades and the costs it faces, and that covariance is recoverable from trajectory data alone. This is the paper's central claim, and three further results follow from it.

\paragraph{The classification is exact and model-free.} For any linear policy, $\operatorname{Cov}(c_t,\hat{\pi}_t(c_t))= \operatorname{tr}(B\Sigma_c)$ (Proposition (ref)): the sign of a single statistic, computed from observed costs and decisions alone, places the strategy exactly into the Kyle1985 dichotomy of liquidity demander or supplier. No order-flow sign, quote data, or knowledge of the strategy's signal is required. Momentum strategies are liquidity consumers and contrarian strategies are liquidity providers as a matter of algebra, not estimation; the only object that must be estimated is the policy matrix $B$ implicit in the observed trajectory, and even that estimation is unnecessary if one only wants the sign.

\paragraph{The same statistic recovers the effective spread.} Under an AR(1) cost process, the auto-covariance correction to the covariance sum equals $\alpha s^2$, the product of strategy size and the squared Roll (1984) implied spread (Theorem (ref)). This is not an analogy or an approximation: it is an exact closed-form identity connecting a performance-attribution statistic to a market-microstructure quantity, derived from the same trajectory data and at no additional computational cost. Empirically, the implied spread recovered this way tracks realized illiquidity through the sample. The implied spread roughly triples during the COVID-19 liquidity crisis and collapses to its sample minimum during the 2022 rate-shock episode, when persistent common-factor repricing dominates short-horizon return dynamics (Section (ref)). The correction is smallest precisely when markets are deep and largest precisely when microstructure breaks down, which is the behavior one wants from an illiquidity proxy.

\paragraph{Aggregate liquidity imbalance produces a closed-form fire sale.} Extending the single-agent result to $N$ correlated strategies under a common-factor cost structure (Section (ref)) shows that the welfare loss from liquidity-balance failure scales as $N^2$ in the strategy-correlation coefficient (Corollary (ref)). When agents' liquidity demand and supply fail to net to zero, prices must adjust to absorb the imbalance. This happens under perfect strategy correlation, when many funds run the same factor-driven allocation. The resulting price-impact regret compounds the covariance regret multiplicatively rather than additively. This gives the familiar fire-sale mechanism a closed form computable from the same observable trajectories used for the single-agent classification, with no model of the deleveraging process itself required.

\paragraph{What the classifier recovers, and what it does not.} The trajectory-based classifier of this paper is deliberately coarser than order-flow-based microstructure measures. The classifier requires only realized costs (asset prices) and position sizes, not signed trades, quote revisions, or limit-order-book depth. The classifier recovers a strategy's average liquidity role over the estimation window rather than its behavior at any single trade. This is a feature when the internal logic of the strategy is unobservable, the setting motivating this paper. When finer-grained order-flow data is available, standard measures Hasbrouck1991 or trade-classification algorithms will recover intraday dynamics this statistic cannot. The two approaches are complements rather than substitutes. The Roll-spread identity of Theorem (ref) gives an explicit bridge between the two approaches, expressing a daily-frequency, trajectory-based quantity in the same units as a standard microstructure spread estimator, so that one can be validated against the other on data where both are available. We view extending this bridge to intraday frequencies, and benchmarking the implied spread directly against effective and realized spread measures computed from TAQ data, as the natural next step for this line of work.

\paragraph{A closing remark on horizon.} The discounted identity of Theorem (ref) shows that a single-period covariance estimate, scaled by the effective horizon $1/(1-\gamma)$, measures long-run discounted regret directly. The same scaling applies to the liquidity classification: a strategy's liquidity role, once identified from a short window of trajectory data, extrapolates to its long-run discounted contribution to market liquidity without re-estimation at every horizon. This makes the single covariance statistic a practical tool not only for one-off classification but also for monitoring how a strategy's liquidity role evolves as market conditions change.